Theresa May has seen the EU reject a key part of her Brexit strategy, with Brussels mounting strong opposition to the prime minister’s customs proposals.

Following the latest round of talks, EU chief negotiator Michel Barnier blew a hole in Mrs May’s post-Brexit plan to collect tariffs on behalf of the bloc at the UK border.

Known as a “facilitated customs arrangement”, the prime minister’s proposal formed a key plank of her plan for leaving the EU, which she agreed with her cabinet at Chequers earlier this month and subsequently set out in a white paper.

The Chequers strategy led Mrs May to suffer two high-profile cabinet resignations – Boris Johnson and David Davis – as well as infuriating her Brexiteer MPs.

Delivering a further blow to her plan, Mr Barnier used a joint news conference with Brexit Secretary Dominic Raab to voice resistance to the customs element of Mrs May’s strategy.

Taking a dig at the Leave campaign’s “taking back control” mantra from the 2016 EU referendum, Mr Barnier warned: “Maintaining control of our money, law and borders also applies to the EU’s customs policy.”

The European Commission official added: “The EU cannot – and will not – delegate the application of its customs policy and rules, VAT and excise duty collection to a non-member, who would not be subject to the EU’s governance structures.”

Calling on the UK to “respect this principle”, Mr Barnier also reiterated that the EU remains open to agreeing a post-Brexit customs union with the UK, despite the prime minister having consistently ruled this out.

He added: “Any customs arrangement will also have to be workable and must protect EU and national revenue, without imposing additional costs on businesses and customs authorities.”

Mr Barnier was also pessimistic about the UK’s proposals on a backstop agreement for the Irish border, which seeks to avoid a hard border on the island of Ireland in the event the issue is not solved by the future EU-UK trading relationship.

The UK has rejected the EU’s own backstop plan, warning it risks dividing Northern Ireland from the rest of the UK.

But, despite the UK’s insistence a backstop agreement must be on a UK-wide basis, Mr Barnier declared Brussels has “doubts that this can be done without putting at risk the integrity of our customs union, our common commercial policy, our regulatory policy, and our fiscal revenue”.

He revealed “open and frank” talks had been held between the EU and UK on the issue, with the UK promising to return with “concrete proposals” on how to address Brussels’ concerns.

Mr Barnier was more positive about the proposals in Mrs May’s Chequers plan for a future security relationship, which he characterised as “a real step forward”.

Responding to his EU counterpart’s remarks, Mr Raab repeated the UK’s demand for an Irish backstop plan to be “time-limited” to prevent an “extended limbo” after Brexit.

He also insisted the UK’s proposals “respect the result” of the Brexit vote, as well as the “core principles of the EU”.

In a message to Brussels, Mr Raab added: “We have considered the innovative approaches the EU has taken in the past with other third countries – when the political will has been there.”

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Mr Raab and Mr Barnier have agreed to meet again in mid-August before holding weekly discussions until October, which has been set as a deadline for concluding a Brexit deal.

The Brexit secretary told Mr Barnier: “Michel, we’ve got work to do.”

Pro-EU campaigners seized on Mr Barnier’s remarks to push their case for a second EU referendum.

Labour MP Ben Bradshaw, a supporter of the People’s Vote campaign, said: “The white paper is dead. It has expired. It has ceased to be. It has gone to join the choir invisible.

“Once again the chaos and confusion at the heart of Brexit is exposed. Mrs May’s scheme could not command a majority in the House of Commons two weeks ago when she allowed the Brextremists to mutilate it.

“Now we know that hardly mattered because it was not acceptable to the EU in any case.”